A Fixed Point Theorem for Two Pair of Maps Satisfying a New Contractive Condition of Integral Type

Transcription

1 International Mathematical Forum, 4, 29, no. 4, A Fixed Point Theorem for Two Pair of Maps Satisfying a New Contractive Condition of Integral Type U. C. Gairola and A. S. Rawat Department of Mathematics Pauri Campus of H.N.B. Garhwal University, Pauri 2461, India ucgairola@rediffmail.com Abstract In this paper we establish a fixed point theorem for two pair of maps satisfying a new contractive condition of integral type by using the concept of occasionally weakly compatible maps. Our result generalize and extend the results of Jha [13], Branciari [6] and many others. Mathematics Subject Classification: 54H25, 47H1. Keywords: Occasionally weakly compatible maps, contractive condition of integral type. 1.INTRODUCTION Prior to 1968 all work involving fixed points used the Banach contraction principle [5]. In 1968 Kannan [16] proved a fixed point theorem for a map satisfying a contractive condition that do not require continuity at each point. This paper was a genesis for a multitude of fixed point papers over the next two decades. (see for instance, [2], [25] for listing and comparison of many of these definitions). In a generalization of Banach contraction principle Meir-Keeler [19] proved the fixed point theorem for (ε, δ)- contractive condition. This result of Meir- Keeler was generalized among others by [7],[12], [14], [15], [21]-[23] and [28]. Recently Jha [13] gave a new generalization of Meir-Keeler type common fixed point theorem stated as follows. THEOREM 1.1 [13]. Let A, B, S and T be self mappings of a metric space (X, d) such that A(X) T (X), B(X) S(X). Assume further that for a given ε > there exists δ > such that for all x, y in X ε < M(x, y) < ε + δ d(ax, By) ε,

2 178 U. C. Gairola and A. S. Rawat where M(x, y) = max{d(sx, T y), d(ax, Sx), d(by, T y), [d(sx, By) + d(ax, T y)]/2} and d(ax, By) < k[d(sx, T y)+d(ax, Sx)+d(By, T y)+d(sx, By)+d(Ax, T y)], for k 1 / 3. If one of the A(X), B(X), S(X) and T (X) is complete subspace of X and if the pair (A, S) and (B, T ) are weakly compatible, then A, B, S and T have unique common fixed point. On the other hand Branciari [6] gave a fixed point result for a single mapping satisfying an analogue of Banach s contraction principle which is stated as follows. THEOREM 1.2 [6]. Let (X, d) be a complete metric space, c [, 1), T : X X a mapping such that, for each x, y X, d(t x,t y) c d(x,y) where f : R + R + is a Lebesgue-integrable mapping which is summable, non-negative and such that, for each s >, s >. Then T has a unique fixed point z X such that, for each x X, lim n T n x = z. This result was further generalized by Abbas and Rhoades [1], Aliouche [4], Gairola and Rawat [8], Kumar et.al. [17] and Rhoades [25]. However, Suzuki [27] has shown that Meir-Keeler contraction of integral type are still Meir-Keeler contraction. In 1996, Jungck ( [9], see also [1] ) introduce the notion of weakly compatible maps and showed that compatible maps are weakly compatible but converse need not be true. Two Maps A and S are said to be weakly compatible if they commute at their coincidence point. Recently Al-Thagafi and Shahzad ( [2], see also [3] ) gave a proper generalization of nontrivial weakly compatible maps which do have a coincidence point. DEFINITION 1.2 ([2], see also [11] ). Two self maps A and S of a set X are said to be occasionally weakly compatible (owc) iff there is a point x in X which is a coincidence point of A and S at which A and S commute. The following example is motivated by example 1.4 of Abbas and Rhoades [1] for a pair of single valued maps which shows that the pair is not weakly compatible but occasionally weakly compatible. EXAMPLE 1.1. Let X = [, ) with usual metric. Define A, S : X X, by {, x < 1 Ax = 2x, 1 x <, Sx = { x, x < x, 1 x <.

3 Fixed point theorem 179 It can be easily verified that x = 1 is coincidence point of A and S, but A and S are not weakly compatible there. However, the pair {A, S} is occasionally weakly compatible (owc). In this paper we prove a common fixed point theorem for two pair of maps by combining the result of Jha [13] and Branciari [6] and also using the concept of owc pair of maps. 2. MAIN RESULT We need the following Lemma before stating our main result. LEMMA 2.1. Let A, B, S and T be self mappings of a metric space (X, d), let f be a summable, non-negative, Lebesgue integrable function from [, ) into itself satisfying s >, for all s > and A(X) T (X), B(X) S(X). Assume further that for given ε > there exists δ > such that for all x, y in X ε < < ε + δ implies ε (2.1) and > implies < (2.2) where M(x, y) = max{d(sx, T y), d(ax, Sx), d(by, T y), [d(sx, By) + d(ax, T y)]/2} (2.3) Then for a fix x in X, the sequence {y n } in X defined by the rule is a Cauchy sequence. y 2n = Ax 2n = T x 2n+1, y 2n+1 = Bx 2n+1 = Sx 2n+2, (2.4) PROOF. Following the proof technique of Jachymski [12] proof is immediate. Now we state our main theorem. THEOREM 2.1. Let A, B, S and T be self maps defined on a metric space (X, d) satisfying the following conditions: A(X) T (X), B(X) S(X), (2.5) given ε > there exists δ > such that for all x, y in X ε < < ε + δ implies ε, (2.6)

4 18 U. C. Gairola and A. S. Rawat for all x, y X, there exists a k [, 1/3] such that [d(sx,t y)+d(ax,sx)+d(by,t y)+d(sx,by)+d(ax,t y)] (2.7) where f is summable, non-negative, Lebesgue integrable function from [, ) into itself satisfying s >, for all s > (2.8) and M(x, y) = max{d(sx, T y), d(ax, Sx), d(by, T y), [d(sx, By) + d(ax, T y)]/2}. If one of the A(X), B(X), S(X) and T (X) is complete subspace of X, then (1) A and S have a coincidence point, or (2) B and T have a coincidence point. Further if the pair {S, A} and {T, B} are occasionally weakly compatible, then (3) A, B, S and T have a unique common fixed point. PROOF. Fix an x in X. By virtue of (2.5) we can define a sequence {y n } in X as y 2n = Ax 2n = T x 2n+1, y 2n+1 = Bx 2n+1 = Sx 2n+2, for all n =, 1,... Suppose d(y 2n, y 2n+1 ) = for some n. Then y 2n = y 2n+1 i.e. Ax 2n = T x 2n+1 = Bx 2n+1 = Sx 2n+2, and T and B have a coincidence point. Similarly, if d(y 2n+1, y 2n+2 ) = for some n. Then Ax 2n+2 = T x 2n+3 = Bx 2n+1 = Sx 2n+2, and A and S have a coincidence point. Further assume that if d(y n, y n+1 ) for each n, then we have M(x, y) >. Otherwise we have d(y n, y n+1 ) =, which is a contradiction. Hence using Lemma 2.1, {y n } is a Cauchy sequence in X. Now, suppose that T (X) is a complete subspace of X. Then the subsequence {y 2n }, contained in T (X) is convergent and has a limit in T (X). Call it u. Let v T 1 u, then T v = u. Similarly the subsequence {y 2n+1 } also converges to u. Now we shall show that Bv = u, Let d(bv, u) >. Then taking x = x 2n and y = v in (2.7) we get d(ax2n,bv) Taking the Limit, as n, [d(sx2n,t v)+d(ax 2n,Sx 2n )+d(bv,t v)+d(sx 2n,Bv)+d(Ax 2n,T v)]. d(u,bv) [d(u,t v)+d(u,u)+d(bv,t v)+d(u,bv)+d(u,t v)] < 2k [d(u,u)+d(u,u)+d(bv,u)+d(u,bv)+d(u,u)] [d(bv,u)+d(u,bv)] d(bv,u)

5 Fixed point theorem 181 which is a contradiction since k [, 1 / 3 ]. Hence from (2.8), Bv = u = T v. Since B(X) S(X), Bv = u implies that u S(X). Then there exists some w in S 1 u, Sw = u. Setting x = w and y = x 2n+1 in (2.7), we get d(aw,bx2n+1 ) Taking the limit, as n, d(aw,u) < 2k [ ] d(sw,t x 2n+1 )+d(aw,sw)+d(bx 2n+1,T x 2n+1 ) +d(sw,bx 2n+1 )+d(aw,t x 2n+1 ) [d(sw,u)+d(aw,u)+d(u,u)+d(sw,u)+d(aw,u)] [d(aw,u)+d(aw,u)] d(aw,u). which is a contradiction. Hence from (2.8), Aw = u = Sw. Hence the sets of the coincidence points of the pairs {S, A} and {T, B} are nonempty and u is unique. This proves (1) and (2). Now to prove (3), note that {S, A} and {T, B} are occasionally weakly compatible and u = T v = Bv = Aw = Sw, then T Bv = BT v and ASw = SAw. If Bu u then from (2.7) d(u,bu) = < k < 3k d(aw,bu) [d(sw,t u)+d(aw,sw)+d(bu,t u)+d(sw,bu)+d(aw,t u)] [d(u,t u)+d(u,u)+d(bu,t u)+d(u,bu)+d(u,t u)] d(u,bu) which is a contradiction for k < 1 / 3. From (2.8), we have Bu = u = T u. Now we will prove that Au = Bu. Using equation (2.7) d(au,bu) [d(su,t u)+d(au,su)+d(bu,t u)+d(su,bu)+d(au,t u)] < 3k d(au,bu) which is a contradiction and hence Au = Bu. Therefore we have u = T u = Bu = Au = Su. The uniqueness of the common fixed point u follows easily from condition (2.7). The same result hold if we assume that S(X) is complete instead of T (X). Now if A(X) is complete then by (2.5), u A(X) T (X). Similarly if B(X) is complete then u B(X) S(X). So the theorem is established.

6 182 U. C. Gairola and A. S. Rawat REMARK 2.1. If we put f(t) 1 in Theorem 2.1, then we have the Theorem 1.1 as a corollary. REMARK 2.2. With the help of an example 2 of Jha [13] shown that his Theorem 1.1 is more general than the previous known results. So our Theorem 2.1 extend the results of Jha-Pant [14], Jha et.al.[15], Pant and Jha [22], Popa [23], Vats [28] and others. ACKNOWLEDGEMENT. The authors thank Professor Naseer Shahzad for his valuable comments. References [1] Mujahid.Abbas and B.E.Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type, Fixed point theory and Application 4 (27) Article ID 5411, 9pages. [2] M.A.Al-Thagafi and Naseer Shahzad, Generalized I-nonexpansive selfmaps and invariant approximations, Acta Mathematica Sinica, English Series 24 (5) (28), [3] M.A.Al-Thagafi and Naseer Shahzad, A note on occasionally weakly compatible maps, Int. J.Math. Anal. 3 (2) (29), [4] A.Aliouche, A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type, J.Math.Anal. Appl. 322 (2) (26), [5] S.Banach, Sur les operations dans les ensembles absraites et leurs applications, Fund.Math. 3 (1922), [6] A.Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int.J.Math.Math.Sci. 29 (9) (22), [7] Lj.B.Ciric, A new fixed point theorem for contractive mappings, Publications de l Institut Mathematique (Beograd) 3 (44) (1981), [8] U.C.Gairola and A.S.Rawat, A fixed point theorem for integral type inequality, Int.J.Math.Anal. 2 (15) 28, [9] G.Junck, Common fixed points for noncontinuous nonself maps on nonmetric spaces, Far East J.Math.Sci. 4 (1996), [1] G.Jungck and B.E.Rhoades, Fixed point for set valued function without continuity, Indian J.Pure Appl. Math. 29 (3) (1998), [11] G.Jungck and B.E.Rhoades, Fixed point theorems for occasionally weakly compatible mappings, J. Fixed Point Theory 7 (2) (26), [12] J.Jachymski, Equivalent conditions and Meir-Keeler type theorems, J.Math.Anal.Appl. 194 (1) (1995), [13] K.Jha, Common fixed point for weakly compatible maps in metric space, Kathmandu Univ.J.Sci.Eng.Appl. I (IV) (27), 1-6. [14] K.Jha and R.P.Pant, A generalization of a Meir-Keeler type common fixed point theorem for compatible maps, Varahmihr J.Math.Sci. 3 (23), [15] K.Jha, R.P.Pant and S.L.Singh, On the existence of common fixed points for compatible mappings, Punjab Univ.J.Math. 37 (25)

7 Fixed point theorem 183 [16] R.Kannan, Some results on fixed points, Bull.Calcutta Math. Soc.6 (1968), [17] S.Kumar, R.Chug and R.Kumar, Fixed point theorem for compatible mappings satisfying a contractive condition of integral type, Sochow J. Math.33 (2) (27), [18] J.Matkowski, Fixed point theorems for contractive mappings in metric spaces, Casopis Pro Pestovant Matematiky, 15 (4) (198), [19] A.Meir and E.Keeler, A theorem on contraction mappings, J.Math.Anal.Appl. 28 (2) (1969), [2] J.Mesjaros, A comparison of various definition of contractive type mappings, Bull.Calcutta Math.Soc.84 (2) (1992), [21] R.P.Pant, Meir-Keeler type fixed point theorems and dynamics of functions, Demonstratio Math. 36 (1) (23), [22] R.P.Pant and K.Jha, A generalization of Meir-Keeler type common fixed point theorem for four mappings, J.Nat.Phys.Sci. 16 (1-2) (22), [23] V.Popa, A generalization of Mier-Keeler type common fixed point theorem for four non-continuous mappings, Sarajevo J.Math. 1 (13) (25), [24] B.E.Rhoades, A comparison of various definitions of contractive mappings, Trans.Amer.Math.Soc. 26 (1977), [25] B.E.Rhoades, Two fixed point theorems for mapping satisfying a general contractive condition of integral type, Int.J.Math.Math.Sci. 23 (63) (23), [26] T.Suzuki, Fixed point theorem for asymptotic contractions of Mier-Keeler type in complete metric spaces, Nonlinear Analysis 64 (26), [27] T.Suzuki, Meir-Keeler contractions of integral type are still Meir-Keeler contractions, Int.J.Math.Math.Sci. 27, Article ID 39281, 6 pages, 27. doi:1.1155/27/ [28] R.K.Vats, Weakly compatible maps in metric spaces, J.Indian Math.Soc.69 (14) (22), [29] P.Vijayaraju, B.E.Rhoades and R.Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, Int.J.Math.Math.Sci. 25 (5) (25), Received: September 26, 28